Sharp Sobolev Sandwich and Approximation Rates of Radon-Domain Lp Ridge Integral Spaces for ReLUk Networks
Abstract
We develop the Lp space and approximation theory for shallow neural networks with ReLUk activations. The central object is the Radon-domain Lp space ℛLpk(Ω) containing all functions on a bounded domain Ω that admit a ridge integral representation whose coefficient density belongs to Lp in the Radon domain. In the Hilbert case p=2, we prove by elementary Fourier analysis that this space recovers the critical Sobolev space Hk+(d+1)/2(Ω). For general 1<p<∞, the identity becomes a sandwich for Bessel-potential Sobolev spaces. The sharp gap of each side is exactly the Seeger--Sogge--Stein loss for the Radon transform as a Fourier integral operator. This also clarifies how the activation regularity and Radon back-projection jointly produce the regularity. As an application, we discretize the integral representation using a deterministic interpolation skeleton plus uniform sampling. This yields high-probability Lp approximation rates and the optimal Hilbert rate O (n−1/2−(2k+1)/2d) at p=2 for linearized neural networks.
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BibTeX
@misc{he2026sharp,
title = {{Sharp Sobolev Sandwich and Approximation Rates of Radon-Domain $L^p$ Ridge Integral Spaces for ReLU$^k$ Networks}},
author = {Juncai He and Zitong Tian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.24795},
eprint = {2606.24795},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.24795},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Sobolev Sandwich and Approximation Rates of Radon-Domain L^p Ridge Integral Spaces for ReLU^k Networks
AU - Juncai He
AU - Zitong Tian
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2606.24795
UR - https://arxiv.org/abs/2606.24795
AB - We develop the L^p space and approximation theory for shallow neural networks with ReLU^k activations. The central object is the Radon-domain L^p space ℛL^p_k(Ω) containing all functions on a bounded domain Ω that admit a ridge integral representation whose coefficient density belongs to L^p in the Radon domain. In the Hilbert case p=2, we prove by elementary Fourier analysis that this space recovers the critical Sobolev space H^k+(d+1)/2(Ω). For general 1<p<∞, the identity becomes a sandwich for Bessel-potential Sobolev spaces. The sharp gap of each side is exactly the Seeger--Sogge--Stein loss for the Radon transform as a Fourier integral operator. This also clarifies how the activation regularity and Radon back-projection jointly produce the regularity. As an application, we discretize the integral representation using a deterministic interpolation skeleton plus uniform sampling. This yields high-probability L^p approximation rates and the optimal Hilbert rate O (n^−1/2−(2k+1)/2d) at p=2 for linearized neural networks.
ER -